Minimal Models of Adapted Neuronal Response to In Vivo–Like ...

Adapting Rate Models 2103
We show that under such conditions, Ermentrout’s model can be easily
generalized, and the adapted response can be obtained as the fixed point of
f = (m − αf, s). For this result to hold, it is necessary that adaptation is
slow compared to the timescale of the neural dynamics. In such a case, the
feedback current αf is a slowly fluctuating variable and does not affect the
value of s. Note that a slow adaptation is a minimal request in the absence
of noise (Ermentrout, 1998).
The proposed model is very general, but it can be used to full advantage
only if the response function is known analytically. This is the case of simple
model neurons, for which the rate function can be calculated, or more
complex neurons whose f-I curve can be fitted by a suitable model function
(e.g., Larkum, Senn, & Lüscher, in press). In section 2, the adapting rate
model is introduced and tested on several versions of IF neurons, whose
rate functions are known and easily computable. The resulting rate models
are checked against the simulations of the full models from which they
are derived, including the leaky IF (LIF) neuron with conductance-based
synaptic inputs. Only slight variations are needed if a different mechanism
of adaptation is considered, as, for example, an adapting threshold for spike
emission, which is dealt with in section 2.2. Evidence is also provided that
the LIF neuron with an adapting threshold is able to fit the response functions
of rat pyramidal neurons (see section 2.3), a result that parallels those
of Rauch et al. (2003) obtained with an afterhyperpolarization current. Finally,
in section 3, we show how the stationary response function can be
used to predict the time-varying activity of a large population of adapting
neurons.
2 Adapting Rate Models in the Presence of Noise
Firing-rate adaptation is a complex phenomenon characterized by several
timescales and affected by different ion currents. At least three phases of
adaptation have been documented in many in vitro preparations, referred to
as initial or one-interspike (ISI) interval adaptation, which affects the first or
at most the first two ISIs (Schwindt, O’Brien, & Crill, 1997), early adaptation,
involving the first few seconds, and late adaptation, shown in response to a
prolonged stimulation (see Table 1 of Sawczuk, Powers, & Binder, 1997, for
references and a list of possible mechanisms).
Initial adaptation depends largely on Ca 2+ -dependent K + current (Sah,
1996; Powers et al., 1999), although other mechanisms may also play a role
(Sawczuk et al., 1997). The early and late phases of adaptation are not well
understood, and several mechanisms have been put forward: in neocortical
neurons, it seems that Na + -dependent K + currents (Schwindt, Spain, & Crill,
1989; Sanchez-Vives, Nowak, & McCormick, 2000), and slow inactivation
of Na + channels (Fleidervish, Friedman, & Gutnik, 1996) may play a major
role; in motoneurons, evidence is accumulating for an interplay between
slow inactivation of Na + channels, which tend to decrease the firing rate,